SOLUTION: Evaluate: What is the sum of all the one�s digits in the following finite sequence? 3^0 ,3^1,3^2 ,3^3 ,...,3^2006

Question 252455: Evaluate: What is the sum of all the one�s digits in the following finite sequence?
3^0 ,3^1,3^2 ,3^3 ,...,3^2006
Answer by drk(1908) (Show Source): You can put this solution on YOUR website!
There is a pattern for the 3's
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81.
3^5 = 243
Notice the pattern {3,9,7,1} repeats after every fourth group. The sum of 3,9,7,and 1 is 20. 2006 / 4 will get you the number of groups. That quotient is 501 remainder 2.
20*501 = 10020. This is the sum of the first 501 groups. Remainder 2 means add 3+9, or 3^2005 + 3^2006 = 3^1 + 3^2 in terms of units digits.
So, 20*501 + 12 = 10032. Don't forget to add back in 3^0 = 1.
The units digit is 2+1 = 3.
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